1.1 The Ginzburg-Landau equations in the London limit

The Ginzburg-Landau (GL) energy is used to describe the behavior of type-II superconductors. In 2D, this energy can be written as

(1.1) $$ \begin{align} \mathrm{GL}_{\varepsilon}(u,A)=\frac12 \int_{{\Omega}} \left(|(\nabla -iA)u|^2+\frac{1}{2{\varepsilon}^2}(1-|u|^2)^2\right) +\frac12 \int_{\mathbb{R}^2} |\mathrm{curl\,} A-h_{\operatorname{\mathrm{\text{ex}}}}|^2. \end{align} $$

Here, ${{\Omega }}\subset \mathbb {R}^2$ is a smooth simply-connected bounded domain, ${\varepsilon }>0$ is a small parameter (the inverse of the GL parameter), $h_{\operatorname {\mathrm {\text {ex}}}}>0$ is another parameter representing the exterior magnetic field, and $A:=(A_1,A_2):{{\Omega }}\rightarrow \mathbb {R}^2$ is the vector-potential of the induced magnetic field which is obtained by $h=\mathrm {curl\,} A:= \partial _1A_2-\partial _2 A_1$ . It is sometimes more convenient to see A as a $1$ -form $A=A_1 dx_1+A_2 dx_2$ in $\mathbb {R}^2$ and h as a $2$ -form $h= d A$ . We will use both points of view in the following. The complex function $u:{{\Omega }}\rightarrow \mathbb {C}$ is called the order parameter. The regions where $|u| \simeq 1$ are in a superconducting phase, whereas the regions where $|u|\simeq 0$ are in a normal phase. The covariant gradient $\nabla _A u=(\nabla -iA)u$ is a vector in $\mathbb {C}^2$ whose coordinates are $ (\partial _1^Au,\partial _2^Au)=(\partial _1 u-iA_1u,\partial _2 u-iA_2u)$ . The limit ${\varepsilon }\rightarrow 0$ corresponds to extreme type-II materials, and this is the regime we consider in this article. Critical points of $\mathrm {GL}_{\varepsilon }$ in the space

(1.2) $$ \begin{align} X:=\{ (u,A)\in H^1({{\Omega}},\mathbb{C})\times H^1_{\text{loc}}(\mathbb{R}^2,\mathbb{R}^2); \mathrm{curl}\, A-h_{\operatorname{\mathrm{\text{ex}}}}\in L^2(\mathbb{R}^2)\} \end{align} $$

are points $ (u,A)\in X$ such that

(1.3) $$ \begin{align} \mathrm{d} \mathrm{GL}_{\varepsilon}(u,A,v,B)&:= \frac{\mathrm{d} }{\mathrm{d} t}\Big|_{t=0}\mathrm{GL}_{\varepsilon}(u+tv,A+tB)=0, \notag\\&\quad\text{ for all} \ (v,B)\in {\mathcal C}^{\infty}(\overline{{{\Omega}}},\mathbb{C})\times {\mathcal C}^{\infty}_c(\mathbb{R}^2,\mathbb{R}^2). \end{align} $$

They satisfy the Euler-Lagrange equations

(1.4) $$ \begin{align} \left\{ \begin{array}{@{}r@{\ }c@{\ }ll} -(\nabla_A)^2u&=&\frac{u}{{\varepsilon}^2}(1-|u|^2) &\text{ in } {{\Omega}} \\ -\nabla^{\perp} h&=&\langle iu,\nabla_A u \rangle & \text{ in } {{\Omega}} \\ h&=& h_{\operatorname{\mathrm{\text{ex}}}} & \text{ in } \mathbb{R}^2\setminus {{\Omega}} \\ \nu \cdot \nabla_A u & =& 0 & \text{ on } \partial {{\Omega}}. \end{array} \right. \end{align} $$

Here, the covariant Laplacian is defined by $ (\nabla _A)^2u=\partial _1^A(\partial _1^Au)+\partial _2^A(\partial _2^Au)$ , and $ \langle iu,\nabla _Au\rangle $ is a vector in $\mathbb {R}^2$ whose coordinates are $ (\langle iu,\partial _1^Au\rangle ,\langle iu,\partial _2^Au\rangle )$ where, for two complex numbers $z,w\in \mathbb {C}$ , we have denoted by $ \langle z,w\rangle $ the quantity $ \frac 12 (z\bar {w}+w\bar {z})$ . We also use the notation $\nabla ^{\perp } h=(-\partial _2h,\partial _1h)$ , and $\nu $ denotes the outward unit normal to $\partial {{\Omega }}$ .

We observe that Equation (1.4) and the energy $\mathrm {GL}_{\varepsilon }$ are invariant under gauge transformations. More precisely, if $(u,A)$ satisfies (1.4), then for any $f \in H^2_{\text {loc}}(\mathbb {R}^2,\mathbb {R})$ , the couple $(ue^{if},A+\nabla f)$ also satisfies (1.4) and $\mathrm {GL}_{\varepsilon }(ue^{if},A+\nabla f)=\mathrm {GL}_{\varepsilon } (u,A)$ . Physically, only the gauge-invariant quantities are relevant; these are, for example, $\mathrm {GL}_{\varepsilon }$ the energy, $|u|$ the local density of superconducting electrons pairs (in the Barden-Cooper-Schrieffer theory), h the induced magnetic field, and $j:=\langle iu,\nabla _Au\rangle $ the current vector. In order to deal with this gauge-invariance, one often works in the so-called Coulomb gauge by requiring

(1.5) $$ \begin{align} \left\{ \begin{array}{@{}r@{\ }c@{\ }ll} \mathrm{div} A &=& 0 & \text{ in } {{\Omega}} \\ A\cdot \nu &=& 0 &\text{ on } \partial {{\Omega}}. \end{array} \right. \end{align} $$

It can be shown that if $ (u,A)$ is in X and satisfies (1.4), and if A is in the Coulomb gauge (1.5), then $ (u,A)\in {\mathcal C}^{\infty }({{\Omega }},\mathbb {C})\times {\mathcal C}^{\infty }({{\Omega }},\mathbb {R}^2)$ , the bound $|u|\leq 1$ holds, and h is in $H^1({{\Omega }})$ ; see [Reference Sandier and Serfaty32, Proposition 3.8, 3.9, 3.10]. Thus, we can replace the fourth equation in (1.4) by

(1.6) $$ \begin{align} h=h_{\operatorname{\mathrm{\text{ex}}}} \text{ on } \partial {{\Omega}}, \end{align} $$

and we can replace the term $\int _{\mathbb {R}^2} |\mathrm {curl\,} A-h_{\operatorname {\mathrm {\text {ex}}}}|^2$ by $\int _{{\Omega }} |\mathrm {curl\,} A-h_{\operatorname {\mathrm {\text {ex}}}}|^2$ if we consider solutions to (1.4).

The behavior of a family of minimizers $ \{(u_{\varepsilon },A_{\varepsilon })\}_{{\varepsilon }>0}$ of $\mathrm {GL}_{\varepsilon }$ in X in the regime ${\varepsilon }\rightarrow 0$ and $\frac {h_{\operatorname {\mathrm {\text {ex}}}}}{|\log {\varepsilon }|}\rightarrow \lambda>0$ has been previously studied in [Reference Sandier and Serfaty29, Reference Sandier and Serfaty32]. The asymptotics of families of general solutions of (1.4) have also been investigated and can be found in [Reference Sandier and Serfaty31, Reference Sandier and Serfaty32]. In this article, we are interested in the behavior of stable critical points of $\mathrm {GL}_{\varepsilon }$ in X. Here, $ (u,A)$ is a stable critical point of $\mathrm {GL}_{\varepsilon }$ in X if $(u,A)$ satisfies (1.4) and

(1.7) $$ \begin{align} \mathrm{d}^2 \mathrm{GL}_{\varepsilon}(u,A,v,B)&:= \frac{\mathrm{d} ^2}{\mathrm{d} t^2}\Big|_{t=0}\mathrm{GL}_{\varepsilon}(u+tv,A+tB)\geq 0,\notag \\& \quad \text{ for all }(v,B)\in {\mathcal C}^{\infty}(\overline{{{\Omega}}},\mathbb{C})\times {\mathcal C}^{\infty}_c(\mathbb{R}^2,\mathbb{R}^2). \end{align} $$

Our aim is to understand if this stability property produces extra conditions in the limit ${\varepsilon } \to 0$ compared to general critical points. Note that local minimizers are stable and thus enter the framework of our study. This question was listed as an open problem (Open problem 15) in [Reference Sandier and Serfaty32]. Before stating our results, we recall briefly some results on global minimizers and general critical points. For $ (u_{\varepsilon },A_{\varepsilon })\in X$ , we set

$$ \begin{align*} j_{\varepsilon}:=\langle iu_{\varepsilon},\nabla_{A_{\varepsilon}}u_{\varepsilon}\rangle \text{ and } \mu (u_{\varepsilon},A_{\varepsilon}):=\mathrm{curl\,} j_{\varepsilon}+\mathrm{curl\,} A_{\varepsilon}. \end{align*} $$

Theorem 1.1 [Reference Sandier and Serfaty32, Theorem 7.2]

Let $ \{(u_{\varepsilon },A_{\varepsilon })\}_{{\varepsilon }>0}$ be a family of minimizers of $\mathrm {GL}_{\varepsilon }$ in X. Assume that $h_{\operatorname {\mathrm {\text {ex}}}}/|\log {\varepsilon }| \rightarrow \lambda $ as ${\varepsilon } \rightarrow 0$ with $0<\lambda <+\infty $ . Then

with $h_*$ the unique minimizer in $\{f\in H^1_1({{\Omega }})=\{f\in H^1({{\Omega }}); \mathrm {tr}_{| \partial {{\Omega }}} f=1\}; \Delta f\in \mathcal {M}({{\Omega }})\}$ of

$$ \begin{align*} E^\lambda(f):=\frac{1}{2\lambda}\int_{{\Omega}} |-\Delta f+f|+\frac12 \int_{{\Omega}} \left( |\nabla f|^2+|f-1|^2\right), \end{align*} $$

and the solution of the obstacle problem

$$ \begin{align*} \left\{\begin{array}{@{}rcll} h_*\in H^1_1({{\Omega}}), \quad h_*\geq 1-\frac{\lambda}{2} \text{ in } {{\Omega}} \\ \forall v\in H^1_1({{\Omega}}) \text{ such that } v \geq 1-\frac{\lambda}{2}, \quad \int_{{\Omega}} (-\Delta h_*+h_*)(v-h_*)\geq 0. \end{array} \right. \end{align*} $$

Furthermore,

$$ \begin{align*} \frac{\mu(u_{\varepsilon},A_{\varepsilon})}{h_{\operatorname{\mathrm{\text{ex}}}}} \rightarrow \mu_* \text{ in } ({\mathcal C}^{0,\gamma}({{\Omega}}))^*, \quad -\Delta h_*+h_*=\mu_*, \end{align*} $$

$$ \begin{align*} \lim_{{\varepsilon}\to 0} \frac{\mathrm{GL}_{\varepsilon}(u_{\varepsilon},A_{\varepsilon})}{h_{\operatorname{\mathrm{\text{ex}}}}^2}=E^\lambda(h_*)=\frac{|\mu_*|({{\Omega}})}{2\lambda}+\frac12 \int_{{\Omega}} \left(|\nabla h_*|^2+|h_*-1|^2\right). \end{align*} $$

This result on minimizers is actually obtained through a $\Gamma $ -convergence result; see [Reference Sandier and Serfaty32, Chapter 7]. Note that the $\Gamma $ -limit is convex and the limiting magnetic field $h_*$ and the limiting vorticity measure $\mu _*$ are uniquely characterized. In particular, whereas global minimizers of $\mathrm {GL}_{\varepsilon }$ may not be unique, their vortices behave in the same way in the mean-field limit. We now turn our attention to critical points of $\mathrm {GL}_{\varepsilon }$ . We make two assumptions which were used in [Reference Sandier and Serfaty31] and then relaxed in [Reference Sandier and Serfaty32, Chapter 13]. In all this article, unless stated otherwise, we assume that $ \{(u_{\varepsilon },A_{\varepsilon })\}_{{\varepsilon }>0}$ is a family in X which satisfies

(1.8) $$ \begin{align} \mathrm{GL}_{\varepsilon}(u_{\varepsilon},A_{\varepsilon})\leq Ch_{\operatorname{\mathrm{\text{ex}}}}^2 \end{align} $$

(1.9) $$ \begin{align} \frac{h_{\operatorname{\mathrm{\text{ex}}}}}{|\log {\varepsilon}|}\rightarrow \lambda \in (0,+\infty) \ (\text{up to a subsequence}), \end{align} $$

where C denotes a constant which is independent of ${\varepsilon }$ . We can then state the following.

Theorem 1.2 ([Reference Sandier and Serfaty31, Theorem 1], [Reference Sandier and Serfaty32, Theorem 1.7 and 13.1])

Let $\{(u_{\varepsilon },A_{\varepsilon })\}_{{\varepsilon }>0}$ be a family of points in X which solve (1.4) with $A_{\varepsilon }$ in the Coulomb gauge (1.5) and such that (1.8)-(1.9) hold. Then, up to extraction of a subsequence,

$$ \begin{align*} \frac{\mu(u_{\varepsilon},A_{\varepsilon})}{h_{\operatorname{\mathrm{\text{ex}}}}} \rightharpoonup \mu \text{ in } \mathcal{M}({{\Omega}}),\quad \quad -\Delta h+h=\mu \text{ in } {{\Omega}} \quad h=1 \text{ on } \partial {{\Omega}}, \end{align*} $$

and

(1.10) $$ \begin{align} \mathrm{div} (T_h)= 0 \text{ in } {{\Omega}} \quad \quad \text{ where } (T_h)_{ij}=\partial_ih \partial_jh -\frac12 (|\nabla h|^2+h^2)\delta_{ij}, \quad \ 1\leq i,j\leq 2.\nonumber\\ \end{align} $$

Here, the divergence of a matrix is a vector whose components are obtained as the divergence of the rows of the matrix. It can be shown – see, for example, [Reference Sandier and Serfaty32, Theorem 13.1] – that with the notation of the previous theorem, $\mu (u_{\varepsilon },A_{\varepsilon })$ is close to a measure of the form $2\pi \sum _{i=1}^{M_{\varepsilon }} d_i^{\varepsilon } \delta _{a_i^{\varepsilon }}$ , where $M_{\varepsilon }\in \mathbb {N}$ , $a_i^{\varepsilon }$ can be thought of as the center of the vortices of $u_{\varepsilon }$ and $d_i^{\varepsilon }\in \mathbb {Z}$ as their degrees. As noted in [Reference Sandier and Serfaty31], Theorem 1.2 is interesting mainly for solutions such that

(1.11) $$ \begin{align} N_{\varepsilon}:=\sum_{i=1}^{M_{\varepsilon}} |d_i^{\varepsilon}| \end{align} $$

is of same order as $h_{\operatorname {\mathrm {\text {ex}}}}$ . If this is not the case, then we should look at the limit of $\mu (u_{\varepsilon },A_{\varepsilon })/N_{\varepsilon }$ instead, but we do not consider this case in this paper. The matrix (or the tensor) $T_h$ is called the stress-energy tensor associated to the energy $\mathcal {L}(h)=\frac 12 \int _{{\Omega }} (|\nabla h|^2+h^2)$ . Equation (1.10) means that h is a stationary point for $\mathcal {L}$ ; that is, that for any vector field $\eta \in {\mathcal C}^{\infty }_c({{\Omega }},\mathbb {R}^2)$ ,

(1.12) $$ \begin{align} \frac{\mathrm{d}}{\mathrm{d} t} \Big|_{t=0} \mathcal{L}(h_t)=0, \text{ with } h_t(x)=h(x+t\eta (x)). \end{align} $$

This condition on h can also be viewed as a criticality condition on the limiting vorticityFootnote ¹ uniquely determining h via $\mu =-\Delta h+h$ in ${{\Omega }}$ and $h=1$ on $\partial {{\Omega }}$ . It is obtained by passing to the limit in the first inner variations (variations of the form (1.12)) of the energy $\mathrm {GL}_{\varepsilon }$ . Although for global minimizers the limiting vorticity $\mu _*$ is absolutely continuous with respect to the Lebesgue measure, the criticality condition (1.10) allows for more singular measures such as measures supported on curves. It was later shown by Aydi in [Reference Aydi5] that some solutions of the GL equations (1.4) satisfying the bounds (1.8)-(1.9) have their vorticity measures that concentrate on lines or on circles.Footnote ² The implications of the condition (1.10) on the regularity of h and $\mu $ were investigated in [Reference Sandier and Serfaty31, Reference Sandier and Serfaty32, Reference Le21, Reference Rodiac27]. In particular, it was obtained in [Reference Rodiac27] that if $h,\mu $ are as in Theorem 1.2, then the absolutely continuous part of $\mu $ is equal to $h\textbf {1}_{\{|\nabla h|=0\}}$ ( $|\nabla h|$ was shown to be a continuous function in [Reference Sandier and Serfaty32, Theorem 13.1]), and the orthogonal part is supported by a locally $\mathcal {H}^1$ rectifiable set. Roughly speaking, it says that $\mu $ can be supported only by sets of nonzero Lebesgue measure or by some curves.

1.2 Main results

In this article, we investigate the following problem: does a stability condition on a family of critical points $ \{(u_{\varepsilon },A_{\varepsilon })\}_{{\varepsilon }>0}$ of $\mathrm {GL}_{\varepsilon }$ in X imply more regularity on their limiting vorticity measures? In particular, can a family of stable solutions of (1.4) have a limiting vorticity which concentrates on curves? To answer this question, our strategy is to pass to the limit in the second inner variation of the GL energy and deduce a supplementary condition for limiting vorticity measures of stable solutions of (1.4). Then we examine if this supplementary condition implies more regularity on $\mu $ .

We first explain more precisely what we mean by first and second inner variations. Let $\eta \in {\mathcal C}_c^{\infty }({{\Omega }},\mathbb {R}^2)$ ; we consider its associated flow map $\Phi :\mathbb {R}\times {{\Omega }} \rightarrow \mathbb {R}^2$ which satisfied that for every $x\in \mathbb {R}^2$ , the map $t\mapsto \Phi (t,x)$ is the unique solution to

(1.13) $$ \begin{align} \left\{ \begin{array}{@{}r@{\ }c@{\ }ll} \frac{\partial }{\partial t}\Phi(t,x)&=&\eta (\Phi(t,x)) \\ \Phi(0,x)&=&x. \end{array} \right. \end{align} $$

It can be seen thanks to the Cauchy-Lipschitz theory that the flow map is well defined in $\mathbb {R} \times {{\Omega }}$ and that it is in ${\mathcal C}^{\infty }(\mathbb {R}\times {{\Omega }})$ . The family $\{\Phi _t\}_{t\in \mathbb {R}}$ with $\Phi _t(\cdot )=\Phi (t,\cdot )$ is a one-parameter group of ${\mathcal C}^{\infty }$ -diffeomorphisms of ${{\Omega }}$ with $\Phi _0=\mathrm {Id}$ . The first and second inner variations of $\mathrm {GL}_{\varepsilon }$ at $ (u,A)$ in the direction $\eta $ are defined by

(1.14) $$ \begin{align} \delta \mathrm{GL}_{\varepsilon}(u,A,\eta )&=\frac{\mathrm{d}}{\mathrm{d} t} \Big|_{t=0} \mathrm{GL}_{\varepsilon}(u\circ \Phi_t^{-1}, (\Phi_t^{-1})^*A), \end{align} $$

(1.15) $$ \begin{align} \delta^2 \mathrm{GL}_{\varepsilon}(u,A,\eta)&=\frac{\mathrm{d} ^2}{\mathrm{d} t^2} \Big|_{t=0} \mathrm{GL}_{\varepsilon}(u\circ \Phi_t^{-1},(\Phi_t^{-1})^* A). \end{align} $$

Here, we have denoted by $ (\Phi _t^{-1})^* A$ the pullback of A, viewed as a 1-form, by the diffeomorphism $\Phi _t^{-1}$ . The reason for taking the pullback $ (\Phi _t^{-1})^* A$ and not only $A\circ \Phi _t^{-1}$ is that we need to respect the gauge invariance. This will be explained in details in Section 2. Note that when working with differential forms, it is customary to take inner variations as pullbacks by $\Phi _t^{-1}$ ; see, for example, [Reference Serre37]. We will also see that (1.14) and (1.15) are well defined and give their expressions in Section 2. In this paper, we do not consider inner variations up to the boundary. This is because we are mainly interested in the regularity of the vorticity measures $\mu $ in the interior. For the use of inner variations up to the boundary in different contexts, we refer, for example, to [Reference Le and Sternberg24, Reference Babdjian, Millot and Rodiac6].

Our main result is the following.

Theorem 1.3 Let $\{(u_{\varepsilon },A_{\varepsilon })\}_{{\varepsilon }>0}$ be a family of points in X which solve (1.4) with $A_{\varepsilon }$ in the Coulomb gauge (1.5) and such that (1.8)-(1.9) hold. Let h be the weak $H^1$ -limit of $h_{\varepsilon }/h_{\operatorname {\mathrm {\text {ex}}}}$ and $\mu $ be the limit in the sense of measure of $\mu (u_{\varepsilon },A_{\varepsilon })/h_{\operatorname {\mathrm {\text {ex}}}}$ given in Theorem 1.2. If we assume that

(H) $$ \begin{align} \lim_{{\varepsilon} \to 0} \frac{\mathrm{GL}_{\varepsilon}(u_{\varepsilon},A_{\varepsilon})}{h_{\operatorname{\mathrm{\text{ex}}}}^2}= \frac{|\mu|({{\Omega}})}{2\lambda } +\frac12 \int_{{\Omega}} \left(|\nabla h|^2+|h-1|^2 \right), \end{align} $$

then either $h\equiv 1$ , or for all $\eta \in {\mathcal C}^{\infty }_c({{\Omega }},\mathbb {R}^2)$ , we have

(1.16) $$ \begin{align} \lim_{{\varepsilon} \to 0} \frac{\delta^2 \mathrm{GL}_{\varepsilon}(u_{\varepsilon},A_{\varepsilon},\eta)}{h_{\operatorname{\mathrm{\text{ex}}}}^2}&= \int_{{\Omega}} \bigg( |\mathrm{D} \eta^T \nabla^{\perp} h|^2-|\nabla h|^2 \det \mathrm{D} \eta+ h^2 [(\mathrm{div} \eta)^2-\det \mathrm{D} \eta]\bigg)\notag \\& \quad +\frac{1}{\lambda}\int_{{\Omega}} \left( \frac{|\mathrm{D} \eta|^2}{2}-\det \mathrm{D} \eta \right) \mathrm{d} |\mu|=:Q_h(\eta). \end{align} $$

If in addition we assume that $\{(u_{\varepsilon },A_{\varepsilon })\}_{{\varepsilon }>0}$ is a family of stable critical points of $\mathrm {GL}_{\varepsilon }$ in X, then h satisfies

(1.17) $$ \begin{align} Q_h(\eta) \geq 0, \quad \text{ for all }\eta \in {\mathcal C}^{\infty}_c({{\Omega}},\mathbb{R}^2). \end{align} $$

It was asked in [Reference Sandier and Serfaty32, Open problem 15] if extra conditions such as (1.7) yield more regularity on the limiting vorticity measure $\mu $ . We have found that the stability condition (1.7) implies (1.17) in the limit. However, we will also see in Proposition 4.1 that a vorticity measure concentrated on a line can satisfy this former property. Since (1.17) is obtained by passing to the limit in the second inner variations, this seems to indicate that stability for inner variations alone is not sufficient to imply regularity (absolute continuity with respect to the Lebesgue measure) on the limiting vorticity measure. Inner variations are usually the variations used to deal with singularities in variational models. However, it still might be possible that, using stability for outer variations genuinely different from inner variations (i.e., not of the form $(\mathrm {D} u_{\varepsilon }.\eta , -\mathrm {D} A_{\varepsilon }.\eta +\mathrm {D} \eta ^T.A_{\varepsilon })$ ), one could obtain further regularity for the limiting vorticity measure.

We now comment on our assumptions (1.8), (1.9), and (H). Assumption (1.8) is quite natural and is satisfied by solutions constructed in [Reference Aydi5, Reference Sandier and Serfaty32, Reference Contreras and Serfaty11, Reference Contreras and Jerrard10]. Assumption (1.9) was used in [Reference Sandier and Serfaty31] in order to have that

(1.18) $$ \begin{align} N_{\varepsilon}\leq C h_{\operatorname{\mathrm{\text{ex}}}}, \end{align} $$

where $N_{\varepsilon }$ is defined in (1.11). Stable solutions of the GL equations (1.4) with an exterior magnetic field much larger than $|\log {\varepsilon }|$ were constructed in [Reference Sandier and Serfaty32, Reference Contreras and Serfaty11, Reference Contreras and Jerrard10], and to this respect, our assumption may appear restrictive. However, (1.9) is satisfied in [Reference Aydi5] where solutions concentrating on lines were built. Hence, singular measures can appear in the limit for this intensity of applied magnetic field, and since our purpose is to study if the stability condition implies some regularity on limiting vorticity measures, it seems natural to consider first this case. We also refer to [Reference Serfaty33, Reference Serfaty34, Reference Serfaty35] and references therein for more results on stable solutions to (1.4). Our main assumption (H) is satisfied by some solutions constructed in [Reference Aydi5]; see Corollary 4.1 and Lemma 4.1 in [Reference Aydi5]. A similar assumption of convergence of energies was used in [Reference Le22, Reference Le23, Reference Le and Sternberg24] to pass to the limit in the second inner variations for the Allen-Cahn problem and also for the nonmagnetic GL problem in dimension bigger than 3 and in a regime where the energy is bounded by $C|\log {\varepsilon }|$ . However, the argument we use is quite different from the ones in the above-mentioned articles which rest upon the use of Reshetnyak’s Theorem for the Allen-Cahn part or on the constancy Theorem for varifolds for the GL part. We note that passing to the limit in the second inner variations for these problems was later shown to be possible without the assumption of convergence of energies in [Reference Gaspar16] and [Reference Cheng9].

1.3 The Ginzburg-Landau equations without magnetic field

We also consider the GL energy without magnetic field

(1.19) $$ \begin{align} E_{\varepsilon}(u)=\frac12 \int_{{\Omega}} \left(|\nabla u|^2+\frac{1}{2{\varepsilon}^2}(1-|u|^2)^2 \right) \end{align} $$

and the associated Euler-Lagrange equation

(1.20) $$ \begin{align} -\Delta u=\frac{u}{{\varepsilon}^2}(1-|u|^2) \text{ in } {{\Omega}}. \end{align} $$

With a fixed boundary condition $g \in {\mathcal C}^1(\partial {{\Omega }},\mathbb {S}^1)$ , this problem has been studied by Bethuel-Brezis-Hélein in [Reference Bethuel, Brezis and Hélein7]. The asymptotic behavior of $\{u_{\varepsilon }\}_{{\varepsilon }>0}$ , solutions to (1.20), depends on the topological degree of g. For a nonzero degree of g, it has been proved in [Reference Bethuel, Brezis and Hélein7] that $ \{u_{\varepsilon }\}_{{\varepsilon }>0}$ converges to some limiting harmonic map with a finite number of singularities (vortices). Besides, vortices of minimizers converge to minimizers of a renormalized energy, vortices of critical points converge to critical points of this renormalized energy, and the stability also passes to the limit as shown in [Reference Serfaty36]. Here, we do not prescribe any boundary condition, and we allow the number of vortices to diverge; however, as in the case with magnetic field, we consider only a family of solutions satisfying the following bound:

(1.21) $$ \begin{align} E_{\varepsilon}(u_{\varepsilon})\leq C |\log {\varepsilon}|^2. \end{align} $$

A way to understand the limit as ${\varepsilon }\to 0$ of solutions $u_{\varepsilon }$ to (1.20) is to look at their phases and at their Jacobian determinants. More precisely, since ${{\Omega }}$ is simply connected and since $\mathrm {div} \langle iu_{\varepsilon },\nabla u_{\varepsilon } \rangle =0$ in ${{\Omega }}$ , then, by using Poincaré’s lemma, we can find $U_{\varepsilon }\in H^1({{\Omega }},\mathbb {R})$ such that

(1.22) $$ \begin{align} \nabla^{\perp} U_{\varepsilon}=\langle iu_{\varepsilon},\nabla u_{\varepsilon}\rangle \text{ in } {{\Omega}} \quad \text{ and } \quad \int_{{\Omega}} U_{\varepsilon}=0. \end{align} $$

Note that $\partial _1 u_{\varepsilon } \wedge \partial _2 u_{\varepsilon }= \mathrm {curl\,} \langle iu_{\varepsilon },\nabla u_{\varepsilon } \rangle =\Delta U_{\varepsilon }$ . Hence, the Laplacian of $U_{\varepsilon }$ is the Jacobian determinant of $u_{\varepsilon }$ , and this quantity was proved to play a prominent role in [Reference Jerrard and Soner19]. We can see here the analogy between $U_{\varepsilon }$ and the magnetic field $h_{\varepsilon }$ in the full GL model. The $\Gamma $ -limit of $E_{\varepsilon }$ in the regime $E_{\varepsilon }(u_{\varepsilon }) \simeq |\log {\varepsilon }|^2$ has been studied in [Reference Jerrard and Soner20], where results analogous to the ones in [Reference Sandier and Serfaty29] are obtained. In particular, in that case, the $\Gamma $ -limit of $E_{\varepsilon }/|\log {\varepsilon }|^2$ is given by $ \frac {|\mu |({{\Omega }})}{2}+\frac 12 \int _{{\Omega }} |\nabla U|^2$ , where U is the weak limit in $H^1$ of $U_{\varepsilon }/|\log {\varepsilon }|$ and $\mu \in \mathcal {M}({{\Omega }})$ is the limit in the sense of measures of $\Delta U_{\varepsilon }/|\log {\varepsilon }|$ (up to extraction). For solutions to (1.20), the following results were obtained in [Reference Sandier and Serfaty31, Reference Sandier and Serfaty32]: the limit satisfies $\Delta U=\mu \in H^{-1}({{\Omega }})$ , and the stress-energy tensor $(S_U)_{ij}=2\partial _iU\partial _jU-|\nabla U|^2 \delta _{ij}$ for $1\leq i,j\leq 2$ is divergence-free in ${{\Omega }}$ . A way to reformulate this property is to say that the quantity $(\partial _xU)^2-(\partial _yU)^2-2i(\partial _xU)(\partial _yU)$ is homomorphic in ${{\Omega }}$ . Using complex analysis techniques and techniques from sets of finite perimeters, it was proved in [Reference Rodiac26] that if $\mu $ satisfies the previous limiting critical conditions, then $\mu $ is supported on a rectifiable set which is locally the the zero set of a harmonic function. Hence, $\mu $ cannot be absolutely continuous with respect to the Lebesgue measure unless $\mu =0$ . We consider here stable solutions of (1.20) – that is, solutions satisfying

(1.23) $$ \begin{align} \frac{\mathrm{d} ^2}{\mathrm{d} t^2}\Big|_{t=0} E_{\varepsilon}(u_{\varepsilon}+tv)\geq 0 \quad \forall v \in {\mathcal C}^{\infty}_c({{\Omega}},\mathbb{C}). \end{align} $$

As we did previously, we define the first and second inner variations of $E_{\varepsilon }$ with respect to a $\eta \in {\mathcal C}^{\infty }_c({{\Omega }}, \mathbb {R}^2)$ by

(1.24) $$ \begin{align} \delta E_{\varepsilon}(u,\eta )=\frac{\mathrm{d}}{\mathrm{d} t} \Big|_{t=0} E_{\varepsilon}(u\circ \Phi_t^{-1}), \quad \quad \delta^2 E_{\varepsilon}(u,\eta)=\frac{\mathrm{d} ^2}{\mathrm{d} t^2} \Big|_{t=0} E_{\varepsilon}(u\circ \Phi_t^{-1}), \end{align} $$

where $\Phi _t$ is defined in (1.13). We will prove in Section 2 that these quantities are well defined.

Theorem 1.4 Let $ \{u_{\varepsilon }\}_{{\varepsilon }>0}$ be a family of solutions to (1.20) satisfying (1.23) and $E_{\varepsilon }(u_{\varepsilon })\leq C|\log {\varepsilon }|^2$ . As shown in [Reference Sandier and Serfaty31, Theorem 3] or [Reference Sandier and Serfaty32], up to a subsequence, $U_{\varepsilon }/|\log {\varepsilon }|\rightharpoonup U$ in $H^1({{\Omega }})$ and $\Delta U_{\varepsilon } / |\log {\varepsilon }|=\mathrm {curl}\, \langle iu_{\varepsilon },\nabla u_{\varepsilon } \rangle /|\log {\varepsilon }| \rightharpoonup \mu \in \mathcal {M}({{\Omega }})$ with

(1.25) $$ \begin{align} \Delta U=\mu \text{ and } (\partial_xU)^2-(\partial_yU)^2-2i(\partial_xU)(\partial_yU) \text{ is holomorphic in } {{\Omega}}. \end{align} $$

If we assume that

(H') $$ \begin{align} \lim_{{\varepsilon} \to 0} \frac{E_{\varepsilon}(u_{\varepsilon})}{|\log {\varepsilon}|^2}= \frac{|\mu|({{\Omega}})}{2}+\frac12 \int_{{\Omega}} |\nabla U|^2 , \end{align} $$

then for all $\eta \in {\mathcal C}^{\infty }_c({{\Omega }},\mathbb {R}^2)$ , we have

(1.26) $$ \begin{align} \lim_{{\varepsilon} \to 0} \frac{\delta^2 E_{\varepsilon} (u_{\varepsilon},\eta)}{|\log {\varepsilon}|^2}&=\frac12 \int_{{\Omega}} |\mathrm{D} \eta ^T\nabla^{\perp} U|^2-|\nabla U|^2 \det \mathrm{D} \eta + \int_{{\Omega}} \left(\frac{|\mathrm{D} \eta|^2}{2}-\det \mathrm{D} \eta \right) \mathrm{d} |\mu| \nonumber\\&=:\tilde{Q}_U (\eta). \end{align} $$

Again, we can see that solutions to (1.20) satisfying (1.23) have the property that, in the limit, $\tilde {Q}_U(\eta )\geq 0$ for all admissible $\eta $ . We can also ask if that condition provides more regularity on the possible limiting vorticies. Here, we obtain that stability for inner variations never implies regularity for the limiting vorticity measure. Indeed, thanks to a recent result of Iwaniec-Onninen [Reference Iwaniec and Onninen18, Theorem 1.12], we are able to prove that every measure satisfying the limiting criticality conditions (1.25) also satisfies the limiting stability condition: $\tilde {Q}_U(\eta )\geq 0$ for all admissible $\eta $ ; see Proposition 4.2. However, we have used only inner variations, and it still might be possible that stability for genuine outer variations could lead to further regularity. We should observe that, contrarily to the case of the GL equations with magnetic field, it is still an open problem to determine if there exist solutions to (1.20) with a diverging number of vortices such that their limiting vorticities concentrate on curves (which should be locally the zero set of some, possibly multi-valued, harmonic functions according to [Reference Rodiac26]).

1.4 Method of proof

For smooth critical points of energies, inner variations are strongly related to outer variations which are defined, in the case of $\mathrm {GL}_{\varepsilon }$ , for $(u,A)\in X$ and $(v,B)\in {\mathcal C}^{\infty }(\overline {{{\Omega }}})\times {\mathcal C}^{\infty }_c(\mathbb {R}^2) $ by (1.3) and (1.7). Although outer variations are of more common use in variational problems, it has been observed that inner variations are useful to understand the limit of singularly perturbed problems such as the Allen-Cahn (AC) problem or the GL problem since these are variations which do move the singularities; see, for example, [Reference Bethuel, Brezis and Hélein7, Reference Hutchinson and Tonegawa17, Reference Lin and Rivière25, Reference Bethuel, Brezis and Orlandi8, Reference Sandier28, Reference Sandier and Serfaty31].

More recently, some interest has grown in understanding how the stability condition passes to the limit in the above-mentioned problems. We refer, for example, to [Reference Le22, Reference Le23, Reference Le and Sternberg24, Reference Gaspar16, Reference Cheng9]. Again, it turns out that studying the second inner variations is more appropriate to understand the limiting behavior of stable solutions to the AC or GL problems. Looking at the expressions given by the first and second inner variations of GL type functionals – see Proposition 2.2 for the formulas – one can see that one of the difficulties is to pass to the limit in quadratic expressions involving derivatives of the unknown functions, whereas only weak convergence in $H^1$ of these functions is available. For example, the vanishing of the first inner variation of $\mathrm {GL}_{\varepsilon }$ provides

(1.27) $$ \begin{align} \mathrm{div}( T_{\varepsilon})=0 \text{ in } {{\Omega}}, \text{ with }(T_{\varepsilon})_{ij} &= \langle \partial_i^{A_{\varepsilon}} u_{\varepsilon},\partial_j^{A_{\varepsilon}}u_{\varepsilon}\rangle\nonumber\\ & \quad -\frac12\left( |\nabla_{A_{\varepsilon}}u_{\varepsilon}|^2+\frac{1}{2{\varepsilon}^2}(1-|u_{\varepsilon}|^2)^2-h_{\varepsilon}^2 \right)\delta_{ij}. \end{align} $$

The formula for the second inner variation is given in Proposition 2.2. Let us briefly recall how Sandier-Serfaty in [Reference Sandier and Serfaty31, Reference Sandier and Serfaty32] managed to pass to the limit in (1.27). First we can see, at least formally, that

(1.28) $$ \begin{align} \frac{T_{\varepsilon}}{h_{\operatorname{\mathrm{\text{ex}}}}^2}\simeq L_{\varepsilon} \text{ where } L_{\varepsilon}=\frac{1}{h_{\operatorname{\mathrm{\text{ex}}}}^2}\left( -\partial_ih_{\varepsilon}\partial_jh_{\varepsilon} +\frac12(|\nabla h_{\varepsilon}|^2+h_{\varepsilon}^2)\right). \end{align} $$

Although $h_{\varepsilon }/h_{\operatorname {\mathrm {\text {ex}}}}$ converges only weakly in $H^1$ , Sandier-Serfaty succeeded in passing to the limit in the equation $\mathrm {div}(T_{\varepsilon })=0$ by showing that the convergence of $h_{\varepsilon }/h_{\operatorname {\mathrm {\text {ex}}}}$ is actually strong in $H^1$ outside a set of arbitrary small perimeter and by using the equation along with a co-area formula argument. This type of problem has the same flavor of the problem of understanding the limit of solutions to the incompressible Euler equations in 2D fluid mechanics; see [Reference DiPerna and Majda13, Reference Delort12].

To pass to the limit in the second inner variation, we cannot use the same argument since we have to pass to the limit in an inequality and not in an equality. We must then understand the limit of all the quadratic terms appearing in the formula given in Proposition 2.2. Assumption (H) allows us to show that the potential term $\frac {1}{2{\varepsilon }^2 h_{\operatorname {\mathrm {\text {ex}}}}^2}(1-|u|^2)^2$ converges strongly toward zero in $L^1({{\Omega }})$ . Next, we say that $|\partial _1^{A_{\varepsilon }}u_{\varepsilon }|^2/h_{\operatorname {\mathrm {\text {ex}}}}^2\rightharpoonup |\partial _2h|^2+\nu _1, \ |\partial _2^{A_{\varepsilon }}u_{\varepsilon }|^2/h_{\operatorname {\mathrm {\text {ex}}}}^2\rightharpoonup |\partial _1h|^2+\nu _2$ and $\langle \partial _1^{A_{\varepsilon }}u_{\varepsilon },\partial _2^{A_{\varepsilon }}u_{\varepsilon }\rangle /h_{\operatorname {\mathrm {\text {ex}}}}^2\rightharpoonup -\partial _1 h,\partial _2h+\nu _3$ , where $\nu _1,\nu _2,\nu _3$ are Radon measures in ${{\Omega }}$ and the convergence takes place in the sense of measures. We can then pass to the limit in the equation $\mathrm {div} ( T_{\varepsilon }/h_{\operatorname {\mathrm {\text {ex}}}}^2)=0$ and use Theorem 1.2 to deduce an equation on $\nu _1,\nu _2,\nu _3$ in the interior ${{\Omega }}$ . This equation actually means that $\nu _1-\nu _2-i\nu _3$ is holomorphic in ${{\Omega }}$ . We use again assumption (H), along with the description of possible limiting vorticity measures $\mu $ obtained in [Reference Rodiac27], to obtain that $\nu _1=\nu _2=\mu /2\lambda $ and $\nu _3=0$ on a ball contained in ${{\Omega }}$ if h is not constantly equal to $1$ . Then the principle of isolated zeros gives $\nu _1=\nu _2=|\mu |/\lambda $ and $\nu _3=0$ in all ${{\Omega }}$ . Finally, we analyze the inequality obtained by passing in the limit in the second inner variation. In the case with magnetic field, we show in one example that we can have $Q_h(\eta )\geq 0$ , where $Q_h$ is defined in (1.16), for all $\eta \in {\mathcal C}^{\infty }_c({{\Omega }}, \mathbb {R}^2)$ and $\mu $ supported by a line. We use similar arguments to treat the case without magnetic field to pass to the limit in the second inner variation. We then employ a result of Iwaniec-Onninen [Reference Iwaniec and Onninen18] to obtain that $\tilde {Q}_U(\eta )\geq 0$ for all $\eta \in {\mathcal C}^{\infty }_c({{\Omega }},\mathbb {R}^2)$ , with $\tilde {Q}_U$ defined in (1.26).

1.5 Organization of the paper

The paper is organized as follows. In Section 2, we compute the expressions of the first and second inner variations. We also explain the link between inner and outer variations. Section 3 is dedicated to show how to pass to the limit in the second inner variation. In order to do this, we study the limit of all the quadratic terms appearing in the second inner variations of $\mathrm {GL}_{\varepsilon }$ by using an argument of defect measures and by using the limit of the first inner variation. Finally, Section 4 is devoted to analyze the limiting stability condition obtained in Theorem 1.3 and Theorem 1.4.

1.6 Notations

For $u,v$ two vectors in $\mathbb {R}^2$ , we denote by $u\cdot v$ their inner product. When $u,v$ are identified with complex numbers, then we denote also their inner products by $\langle u, v \rangle $ . If $\eta \in {\mathcal C}^{\infty }({{\Omega }},\mathbb {R}^2)$ is a smooth vector field, we use $\mathrm {D} \eta $ to denote its differential. When we apply this differential to a vector $x\in \mathbb {R}^2$ , we use $\mathrm {D} \eta .x$ . The second derivative of a smooth vector field $\eta $ applied to two vectors $x,y\in \mathbb {R}^2$ is denoted by $ \mathrm {D}^2 \eta [x,y]$ . For two matrices $M,N \in \mathcal {M}_2(\mathbb {R})$ , we let $ M:N:=\mathrm {tr} (M^TN)$ denote their inner product and $\|M\|$ the associated norm, with $M^T$ the transpose matrix of M. For two vectors $x,y\in \mathbb {R}^2$ , we define their tensor products to be a matrix in $\mathcal {M}_2(\mathbb {R})$ whose entries are given by $ (x \otimes y)_{ij}=x_iy_j$ . Note that we have the relation $M.x \cdot y=M: y\otimes x$ . For $0$ and $1$ -forms f and A, we denote by $df$ and $ dA$ their exterior derivatives. For a function h regular enough. we set $\nabla ^{\perp } h= (-\partial _2h,\partial _1h)^T$ . For a Radon measure $\mu \in \mathcal {M}({{\Omega }})$ , we denote by $|\mu |({{\Omega }})$ its total variation. When we need to evaluate the energy on a subdomain $V\subset {{\Omega }}$ we write $\mathrm {GL}_{\varepsilon }(u,A,V)$ .